Negative Masses, Ma jorana Neutrinos, and the Seesaw Mechanism in the Geometric Algebra Cl(10, 2)

We examine the quantum-mechanical foundations of negative-mass states inCliord-algebraic frameworks. Building on the observation of Debergh and Petit thata unitary time-reversal operator leads to negative-energy solutions of the Dirac equa-tion, and on their use of complexied spacetime algebra to classify four types of mat-ter, we propose an alternative formulation in the real geometric algebra Cl(4, 2). Thefour matter sectorspositive mass/positive charge, positive mass/negative charge,negative mass/positive charge, negative mass/negative chargeemerge from projec-tions onto the null basis vectors of the conformal space, avoiding complexicationentirely. The unitary PT-operator is reinterpreted as the conformal inversion inCl(4, 2).The main result concerns the extension to Cl(10, 2) = Cl(4, 2) ⊗ Cl(6). We iden-tify 36 mixed bivectors that couple the conformal (mass-sign) and internal (avor)sectors. These operators provide a geometric origin for the PMNS neutrino mixingmatrix: avor oscillations correspond to rotations in the joint conformal-internalspace. The seesaw mechanism acquires a geometric interpretation: the heavy right-handed neutrino is the projection of the neutrino eld onto the negative-mass sector,and the Majorana condition ψ = ψ c [16] selects precisely those fermionscharge-neutral neutrinosthat can oscillate between the two mass sectors. The neutrinomass scale is linked to the cosmological sector ratio ξ = a(−) /a(+) through the samedilaton constraint that governs bimetric cosmology. The framework predicts thatneutrinos must be Majorana particles, testable by neutrinoless double beta decayexperiments.